[alma-config]multiplication of synthesized beam by primary beam in optimization

[Bryan Butler]

all,

keep in mind that leonia has *never* been arguing about the
mosaicing case - only the *single-field* case.  he has been
trying, unsuccessfully, to make this point for 6 months.

if we say simply that the compact configuration will do mostly
mosaicing (although i suspect it will be less than the
86.3% number that mark puts forward, i think it is safe to
say that it will spend a majority of time mosaicing), then
the argument is non-existent, since mark has shown (and dave
has reproduced/verified) that in that case you multiply by
the PB*PB.  so, for the compact configuration, this is the
thing to do.

the intermediates are less clear, since i would not be willing
to argue that any of them (aside from the most compact few) will
spend more than 50% of their time mosaicing.  in this case, you
cannot use the mosaicing argument, and must develop an argument
for single-field imaging.  dave has done this, but i myself do not
follow his argument (apologies to dave).  i'm also uncomfortable
with a theory that has such an easily demonstrable counter-example.
yes, it is a statistical argument, but, again, it makes me uncomfortable.
i made an argument similar to steve's argument below in:
http://listmgr.cv.nrao.edu/pipermail/alma-config/2002-January/000490.html
i.e., i agree with steve that i think it is more complicated
than simply stating that the thing to optimize is the SB multiplied
by the PB (or the sqrt(PB), or the PB*PB, or whatever function of
the PB you want to pick).  i also agree with steve that it probably
won't matter much in the end, since with 64 antennas you get such
good coverage that the differences between configs will probably
be transparent (i.e., imaging errors will *not* depend on the
differences between antenna placement, but rather on calibration
errors [including systematics]).

i defer to dave & mark on this - they understand it better than i do.

i would like to see john's comments on this topic though.


	-bryan

p.s. great news from europe! :)...



On 2002.07.08 16:54 Mark Holdaway wrote:
> 
> 
> I have been trying, unsuccesfully, to make this point for 4 years.
> (Does anyone really read these messages?)
> Here goes again.
> (I basically agree with Steve.)
> 
> > The standard "synthesized beam" SB = FT of (weighted) sampling function 
> > (see imaging lecture in book) is what we normally use as the PSF.  We
> > wish to use a mock version of this in our optimization of configurations,
> > e.g. minimize the maximum sidelobe over some area of the sky.  Note that
> > the sampling used is either a bunch of delta-functions (for a DFT image)
> > or gridded cells with weights (what we really use in FFT imaging).  I 
> > assume natural weighting is what is used by the optimizers (but see 
> > below).
> > 
> > Multiplication by the primary beam PB is effectively a penalty function
> > on the SB, downweighting the far-out sidelobes with respect to the inner
> > sidelobes in the optimization.  This makes some sense, in that I would 
> > probably rather have a 5% sidelobe further out than near the main lobe,
> > given the choice.  However, using the PB as the penalty function 
> > introduces some peculiarities --- for example, at the nulls of the PB,
> > the sidelobes will not count in the optimization, which does not make
> > alot of sense.
> 
> (my argument for PB*PB)
> 
> Exactly.  Lets say we have a beam which is 20 arcseconds from center
> to null, and we optimized the PSF multiplied by the primary beam, and
> we got MONSTER (10%) sidelobes right at the first null.  NOW, also imagine
> that we have a 1 Jy source 10 arcs to the RIGHT of the pointing center.
> That monster sidelobe might show up now 10 arcs to the LEFT of the
> pointing center ==> THIS IS BAD.  In fact, we could have a source 19.9
> arcs to the RIGHT, and need to worry about sidelobes 19.9 arcs to the
> LEFT, so we need to make considrations of +/- 40 arcs for optimization.
> SO, just as in CLEAN where the dirty beam needs to be twice as big as the
> model image, we need to optimize over a region twice as large (area, 4x)
> as the primary beam.
> 
> But, you might argue, that 1 Jy source at the half power point actually
> only shows up as a 0.5 Jy source, so the sidelobe halfway across the
> image is .1 * 0.5 Jy, or 0.05 Jy.  In addition, if you are MOSAICING,
> you will multiply the DIRTY IMAGE by the primary beam prior to adding
> to the other pointings' dirty images, so that killer sidelobe is now
> at the 0.025 Jy level.  This seems to be saying, then, that a sidelobe
> 20 arcs off the center is only 25% as important as a sidelobe near the
> center of the PSF.  You can do this analysis again for a 1 Jy source
> at the beam center, and as this 20 arcs-distant sidelobe is at a PB null,
> you figure it is like 0% as important as a sidelobe near the PSF center.
> Since we are mosaicing, and bright emission can be anywhere in the beam,
> we need to integrate over the full beam, putting that 1 Jy source
> everywhere, and the answer ends up looking like PB * PB.
> 
> 
> > Note, I guess when Dave and Leonia talk about the PB, they use a Gaussian 
> > approximation to the primary beam (GPB), which has better behavior in this 
> > regard, but doesn't make much sense physically.  In this case, it is just 
> > a penalty function (monotonically decreasing with distance) with an inner 
> > core approximating the inner core of the PB.
> > 
> > Dave argues based on (I have to admit a rather confusing to me) a 
> > definition of the "image" as reradiation of a perfectly phased voltage
> > pattern from the feeds back through the antennas and the array, which has
> > the PB in it.  Personally, I find this description incomplete as it does
> > not deal with deconvolution (the real issue here, I think).  However,
> > Dave does also point out that if you use actual delta-functions in the 
> > uv-plane as the "visibility" sampling the sidelobes go on forever, which
> > is clearly silly.  A visibility has support over the cross-correlation
> > between the two aperture (voltage) illumination patterns, so there is
> > a built-in scale there on the sky (the FT of this, which I think is what
> > we are calling the PB).  However spreading the sampling function over this
> > uv kernel isn't exactly right either (in particular, mosaicing changes 
> > this part), but I would need some more time to look into this.
> > 
> > Leonia (correctly) points out that the effect of the sidelobes on 
> > reconstruction of emission depends on the location of the emission in the
> > PB, and thus does not like to use the PB for a general case.  I would say
> > that using the bare SB is the most "conservative" case in this regard, 
> > though again I worry a little about the trade of inner for outer 
> > sidelobes.
> > 
> > I think one compromise we can use if we want a penalty function is the 
> > auto-correlation of the PB (= PB*PB).  Dave sort of mentions this in
> > one of his emails.  I think this is what you get if you consider 
> > mosaicing, for example, dealing with the interaction between different
> > pointings.  It is sqrt(2) "wider" in the core than the PB 
> > itself, and has somewhat better behavior (I think). Maybe a Gaussian 
> > GPB*GPB = sqrt(2) wider than the GPB will work.  This sort of considers 
> > the mean square effect of sources distributed over the primary beam 
> > interacting with the sidelobes of each other. 
> > 
> > OK, here is my take on this.  We can use:
> > 
> > 1) the "bare" synthesized beam (SB) over some field of view (e.g. N 
> > primary beamwidths) - this tries to dampen the far-out sidelobes at the
> > cost of inner sidelobes.  This is probably most appropriate for small
> > sources which can be all over the sidelobes of the beam.  Note that at 
> > low frequencies with the EVLA the large number of confusing sources far 
> > out in the beams would lead me to choose this as a "conservative" choice.  
> > Might also be ok for ALMA cases where the source is much larger than 
> > the PB (extreme mosaicing) with lots of bright emission further out, but
> > I think you would be better off using the correct mosaic primary beam 
> > here.
> > 
> > 2) the SBxPB (or SBxGPB, the Gaussian version), which will downweight the
> > outer parts of the beam in the optimization, so you will be minimizing the
> > near-in sidelobe levels.  This would probably be the best choice for 
> > observing sources smaller than the PB (e.g. EVLA NMA & VLBA).
> 
> Maybe OK for the large arrays.  
> 
> > 3) the SBx(PB*PB) or SBx(GPB*GPB), not really much wider than #2, but does
> > push out a bit more into the beam.  I think this is a better choice when
> > considering sources around the PB size or a few times larger - and I 
> > postulate that this approximates the mosaic synthesized beam.  I would say 
> > this is the normal ALMA case.
> 
> This is what I have argued for, intermittantly, for 4 years.
> 
> > In the interest of moving this along, I would adopt #3 for ALMA, I think
> > in the end the different optimized configs no matter what is chosen will
> > be more than acceptable, and this at least tries to take the mosaicing
> > into account.
> 
> Choose this for configurations which will spend a lot of time mosaicing,
> where you have bright emission everywhere in the beam.
> 
> > Note - for an E-config for the EVLA, I would probably also adopt #3, as 
> > you are probably most interested in using this configuration for 
> > mosaicing.
> > 
> > Hope this helps and not further confuses...
> > 
> >   -Steve
> > 
> 
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